Learning Abstract Algebra with ISETL

Springer New York Berlill Heidelberg Bareelolla Budapest Hong Kong London Mila II Paris Santa Clara Singapore Tokyo

Ed Dubinsky Uri Leron

Learning Abstract Algebra with ISETL DOS Diskette Provided

, Springer

Ed Dubinsky Departments of Curriculum &

Instruction and Mathematics Purdue University West Lafayette, IN 47907 USA

With 2 Illustrations

Uri Leron Department of Science Education Technion Israel Institute of Technology 32000 Haifa Israel

Mathematics Subject Classification (1991): 13-01,20-01

Library of Congress Cataloging-in-Publication Data Dubinsky, Ed.

Learning abstract algebra with ISETL / Ed Dubinsky, Uri Leron. p. cm.

Includes bibliographical references and index. Learning abstract algebra with ISETL / Ed Dubinsky, Uri Leron.

Additional Material to this bookcan be downloaded from http://extras.springer.com

ISBN-13:978-1-4612-7602-9 e- ISBN-13:978-1- 4612-2602-4 D01:10.1007/978-1-4612-2602-4

I. Algebra, Abstract - Computer-assisted instruction. 2. ISETL (Computer program language) I. Leron, Uri. II. Title. QA162.D83 1993 S12'.02'078-dc20 93-2609

Printed on acid-free paper.

© 1994 Springer-Verlag New York, Inc. Reprint of the original edition < 1994 >

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereaf­ter developed is forbidden. The programming language ISETL on the enclosed diskette is copyrighted by Gary Levin. It is being jistributed herewith by permission of Gary Levin. The utility programs on the enclosed diskette are under copyright protection. Copying the enclosed diskette for the purpose of making a profit is forbidden. Before using the programs please consult the technical manuals provided by the manufacture.r of the computer. The use of general descriptive m:.mes, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by Natalie Johnson; manufacturing supervised by Vincent Scelta. Camera-ready copy prepared using the authors' LaTeX files.

9 8 7 6 5 4 3 2 (Corrected second printing. 1998)

ISBN-I3:978-1-4612-7602-9 SPIN 10662896

Contents

Comments for the Student

Comments for the Instructor

Acknowledgments

1 Mathematical Constructions in ISETL 1.1 Using ISETL . . . . .

1.1.1 Activities............. 1.1.2 Getting started ......... . 1.1.3 Simple objects and operations on them 1.1.4 Control statements . . . . . . . . . . 1.1. 5 Exercises .... . . . . . . . . . . .

1.2 Compound objects and operations on them 1.2.1 Activities 1.2.2 TUples.... . .... 1.2.3 Sets ......... . 1.2.4 1.2.5 1.2.6 1.2.7 1.2.8 1.2.9

Set and tuple formers Set operations Permutations . . . . . Quantification ... . Miscellaneous ISETL features VISETL

1.2.10 Exercises ........... .

xi

xvii

xxi

1 1 1 5 6 7 8

11 11 14 15 16 17 17 18 20 20 21

vi Contents

1.3 Functions in ISETL 1.3.1 Activities . . 1.3.2 Funcs .. . . 1.3.3 Alternative syntax for funcs . 1.3.4 Using funcs to represent situations 1.3.5 Funcs for binary operations 1.3.6 Funcs to test properties 1.3.7 Smaps. . 1.3.8 Procs . . 1.3.9 Exercises

2 Groups 2.1 Getting acquainted with groups.

2.2

2.3

2.1.1 Activities ..... . 2.1.2 Definition of a group. 2.1.3 Examples of groups .

Number systems. Integers mod n . Symmetric groups. Symmetries of the square . Groups of matrices ....

2.1.4 Elementary properties of groups 2.1.5 Exercises . . .......... . The modular groups and the symmetric groups 2.2.1 Activities ..... . .. . 2.2.2 The modular groups Zn . 2.2.3 The symmetric groups Sn

Orbits and cycles 2.2.4 Exercises . . Properties of groups .... . 2.3 .1 Activities ...... . 2.3.2 The specific and the general . 2.3.3 The cancellation law-An illustration of the

abstract method . . . . . . . . . 2.3.4

2.3.5 2.3.6 2.3.7

How many groups are there? .. Classifying groups of order 4

Looking ahead-subgroups .,. Summary of examples and non-examples of groups Exercises

3 Subgroups 3.1 Definitions and examples.

3.1.1 Activities. . ... 3.1.2 Subsets of a group

Definition of a subgroup

25 25 31 32 33 33 33 34 35 35

39 39 39 42 43 43 45 47 49 52 52 55 57 57 60 65 68 69 71 71 72

74 75 77 79 80 81

83 83 83 86 86

Contents vii

3.1.3 Examples of subgroups. . . . . . . . . 88 Embedding one group in another 88 Conjugates . . . . . . . . . . . . . 89 Cycle decomposition and conjugates in Sn 91

3.1.4 Exercises ...... . . . . 92 3.2 Cyclic groups and their subgroups .......... 94

3.2.1 Activities............... .... 94 3.2.2 The subgroup generated by a single element. 96 3.2.3 Cyclic groups . . . . . 100

The idea of the proof 101 3.2.4 Generators .... .. 103

Generators of Sn 103 Parity-even and odd permutations . 104 Determining the parity of a permutation. . 105

3.2.5 Exercises .. 105 3.3 Lagrange's theorem. . . . . . . . . . . . . . . 108

3.3.1 Activities................ 108 3.3.2 What Lagrange's theorem is all about 111 3.3.3 Cosets............... 112 3.3.4 The proof of Lagrange's theorem 113 3.3.5 Exercises ............ 116

4 The Fundamental Homomorphism Theorem 4.1 Quotient groups .... .

4.1.1 Activities... ........... . 4.1.2 Normal subgroups .......... .

Multiplying cosets by representatives 4.1.3 The quotient group. 4.1.4 Exercises

4.2 Homomorphisms .. .. 4.2.1 Activities . .. .. 4.2.2 Homomorphisms and kernels 4.2.3 Examples........... 4.2.4 Invariants........ .. 4.2.5 Homomorphisms and normal subgroups

An interesting example 4.2.6 Isomorphisms. 4.2.7 Identifications .... . 4.2.8 Exercises ...... .

4.3 The homomorphism theorem 4.3.1 Activities....... 4.3.2 The canonical homomorphism. 4.3.3 The fundamental homomorphism theorem 4.3.4 Exercises . . . . . . . ........ .

119 119 119 121 124 125 126 129 129 133 133 135 136 137 138 139 141 143 143 145 147 150

viii Contents

5 Rings 5.1 Rings

5.2

5.3

5.1.1 5.1.2 5.1.3 5.1.4

5.1.5 5.1.6 5.1.7 5.1.8 5.1.9 Ideals 5.2.1 5.2.2 5.2.3

5.2.4

5.2.5

Activities . . . . . . Definition of a ring . Examples of rings Rings with additional properties

Integral domains ... . . . Fields . . . .. ...... . .

Constructing new rings from old-matrices Constructing new rings from old-polynomials Constructing new rings from old-functions Elementary properties-arithmetic Exercises

Activities Analogies between groups and rings Subrings ......... .

Definition of subring Examples of subrings. . . .

Subrings of Zn and Z . Subrings of M(R) ... Subrings of polynomial rings Subrings of rings of functions

Ideals and quotient rings. Definition of ideal . . . . . Examples of ideals ....

5.2.6 Elementary properties of ideals 5.2.7 Elementary properties of quotient rings

Quotient rings that are integral domains­prime ideals . . . . . . . . . . . . . . . . . Quotient rings that are fields-maximal ideals

5.2.8 Exercises .......... . Homomorphisms and isomorphisms . . . . . . . . . . . 5.3.1 Activities . .... . ......... .. .... 5.3.2 Definition of homomorphism and isomorphism

Group homomorphisms vs. ring homomorphisms 5.3.3 Examples of homomorphisms and isomorphisms.

Homomorphisms from Zn to Zk . . . Homomorphisms of Z . . . . . . . . . . . . . Homomorphisms of polynomial rings . . . . Embeddings-Z, Zn as universal subobjects The characteristic of an integral domain and a field .... . ... .

5.3.4 Properties of homorphisms Preservation . . . . . .

153 153 153 156 156 157 157 158 159 161 164 165 165 168 168 170 171 171 171 171 172 172 173 173 173 175 175 176

176 177 178 181 181 182 183 183 183 184 184 184

185 186 186

Contents ix

Ideals and kernels of ring homomorphisms 186 5.3.5 The fundamental homomorphism theorem 187

The canonical homomorphism . 187 The fundamental theorem . . . 187 Homomorphic images of Z, Zn 188 Identification of quotient rings 188

5.3.6 Exercises ... . .. . .. 190

6 Factorization in Integral Domains 193 6.1 Divisibility properties of integers and polynomials. 193

6.1.1 Activities . . . . . . . . . . . . 193 6.1.2 The integral domains Z, Q[x] . . . . . . . 198

Arithmetic and factoring . . . . . . . 198 The meaning of unique factorization 199

6.1.3 Arithmetic of polynomials. . . . . 200 Long division of polynomials . . . . . 200

6.1.4 Division with remainder . . . . . . . . . . 202 6.1.5 Greatest Common Divisors and the Euclidean

algorithm . . . . . . . . . . . . . . . 204 6.1.6 Exercises . . . . . . . . . . . . . . . 208

6.2 Euclidean domains and unique factorization 209 6.2.1 Activities........... 209 6.2.2 Gaussian integers. . . . . . . . . . . 212 6.2.3 Can unique factorization fail? . . . . 214 6.2.4 Elementary properties of integral domains 214 6.2.5 Euclidean domains . . . . . . . . . . . . . 218

Examples of Euclidean domains . . . 219 6.2.6 Unique factorization in Euclidean domains . 221 6.2 .7 Exercises .. ... .. . . . . 225

6.3 The ring of polynomials over a field. 226 6.3.1 Unique factorization in F[x] . 227 6.3.2 Roots of polynomials. . . . . 228 6.3 .3 The evaluation homomorphism 230 6.3.4 Reducible and irreducible polynomials 231

Examples. . . . . . . . . . . . . . 231 6.3 .5 Extension fields . . . . . . . . . . . . . 235

6.3.6 6.3.7

Index

Construction of the complex numbers . Splitting fields Exercises ..... . ............ .

237 237 239

241

Comments for the Student

Working to learn mathematics

This book is very different from other mathematical textbooks you have used in the past. Using any textbook represents an implicit "contract" between the writer and the reader, between the instructor and the student. In our "contract" we ask more from you and in turn we promise you will get more.

We ask you to work hard. And we really mean work. You will not be a passive recipient of mathematical knowledge. Rather, you'll be actively invohed in doing things (mainly on the computer) and in discussing them with your fellow students. In fact, you and your colleagues will be, collec­tively, constructing your communal and personal knowledge. So when we say "work", we mean more than just putting in hours and doing the assign­ments. We are asking you to THINK - not only about the mathematics, but about what is going on in your mind as you try to learn Abstract Algebra. We are, in fact, asking you to take into your own hands the re­sponsibility for your own learning and that of your colleagues.

And in return, we promise you that this book will make it possible for you to learn the mathematics meaningfully. If you use it properly and if your course is consistent with the principles on which this book is written, then you will begin to own your mathematical ideas, you will become an active learner, and you will learn a great deal of mathematics. You will see that abstract mathematical concepts start to make sense to you. You will be able to understand these concepts and even succeed in proving things

xii Comments for the Student

about them. Less often will you find yourself stuck, staring at the symbols as if they were just so many ink stains on the paper. Indeed, we really can promise you that the symbols of mathematics will have meaning for you because, in working with the computer and reflecting together with your colleagues on that work, you will have constructed the meanings that the symbols represent. You will begin to control mathematics, rather than be controlled by it.

And perhaps we can promise you one more thing. If you are successful in this course, you will begin to see some of the beauty of mathematics. You will become initiated to a way of thinking that goes beyond utility and has for centuries fascinated some of the most powerful minds that the human species has produced. You will touch a little bit these minds and stand on some very tall shoulders. We think you will be very pleased with what you see.

Constructing mathematical ideas

In using this book to learn abstract algebra, much of what you will be do­ing is constructing mathematical ideas on a computer. You will write small pieces of code, or "programs" that get the computer to perform various mathematical operations. In getting the computer to work the mathemat­ics, you will more or less automatically learn how the mathematics works! Anytime you construct something on a computer then, whether you know it or not, you are constructing something in your mind.

This is a fairly new approach to learning mathematics, but it has proved effective in a number of courses (including abstract algebra) over a period of several years. It works a lot better if you get into the spirit of things. In order to learn mathematics you have to somehow figure out most ideas for yourself. This means that you will often find yourself in situations where you are asked to work a problem without ever having been given an explanation of how to do it. We do this intentionally. The idea is to get you to think for yourself, to make the mathematical ideas your own by constructing them yourself, not by listening to someone talk about them.

A lot of people in this situation have a natural question. "If I am asked to solve a problem, or do something on the computer, and I am not told how to do it, what should I do?" The answer is very simple. Make some guesses (better yet, conjectures) and try them out on the computer. Ask yourself if it worked - or what part of your guess worked and what part did not. Try to explain why. Then refine your guess and try again. And again. Keep repeating this cycle until you understand what is going on. The most important thing for you to remember is not to think of these explorations in terms of success and failure. Whenever the computer result is different from what you expected, think of this as an opportunity for you to improve your understanding. Remember: instead of just being stuck, not

Comments for the Student xiii

knowing what to do next, you now have an opportunity to experiment, to make conjectures and try them out, and to gradually refine your conjectures until you are satisfied with your understanding of the topic at hand.

Another piece of advice is to talk about your work. Talk about your problems, your solutions, your guesses, your successes and your failures. Talk to your fellow students, your teachers, to anyone. It is not enough in mathematics to do something right . You have to know what you are doing and why it was right . Talking about what you are doing is a good way of trying to understand what you did. When we teach this course, our students always work in teams, whether they are doing computer activities or paper and pencil homework assignments. And they always talk about what they are doing.

Executing mathematical expressions on the computer and in your mind

Writing definitions and proofs and solving mathematical problems is like writing programs in a mathematical programming language and executing them in your head. Most people find this very hard to do. In this book we offer you a way to have the computer help in executing these mathematical expressions, so that you can always test your conjectures and compare your expectations with the results on the computer screen. Our experience shows that, as students become better at carrying out mathematical activities on the computer, their ability to "run" them in their mind improves as well.

Your computer work will be with the programming language ISETL. The nice thing about this language is that the way it works is very close to the way mathematics works. When you have written some ISETL code and you are running it, try to think about what the computer is doing and how it manipulates the objects you have given it. When you do this, you are actually figuring out how some piece of mathematics works. Another nice thing about this language is that learning to program in it is very sim­ilar to learning the mathematics involved. There's very little programming "overhead" .

If you are not already familar with the computer system that you will be using, then you should expect to spend some hours practicing. If you are really new to this sort of thing, then, at first, it will seem very strange to you and things might go very slowly. Don't be discouraged. Everybody starts slowly in working with computers, but things get better very quickly.

xiv Comments for the Student

Learning with this book

The book consists of six chapters and each chapter is divided into sections (usually three or four). The structure of individual sections in this book reflects our beliefs (supported by contemporary theory and research) on how people learn best. We believe (together with Dewey, Montessori, Pa­pert, and Piaget) that people learn best by doing and by thinking about what they do. The abstract and the formal should be firmly grounded in experience.

Thus each section starts with a substantial list of activities, to be done in teams working on a computer. These are intended to create the experiential basis for the next learning stages. The activities in each section are followed by discussions, introducing the "official" subject matter. Some of these dis­cussions read like a standard mathematics book with definitions, examples, theorems and proofs. But there are important differences. Explanations in the text are often only partial, raising more questions than they answer. Sometimes, an issue is left hanging and only you, the reader, can supply the missing link. Sometimes the discussion even uses ideas which will only be discussed officially some pages later. Once again, this style represents our realization that mathematical ideas can not be given to you. You must make them yourself. All we can do is try to create situations in which you are likely to construct appropriate mathematical ideas.

One reason why this kind of discussion works is because in reading the text you are not being introduced to totally unfamiliar material. Rather, it is just a more general and formal summary of what you have previously experienced and talked about in doing the activities. It summarizes and generalizes your own experiences. It is very important to remember that the activities are meant for doing, not just reading, and for doing before reading the discussions in the text (however, you will not be penalized if you decide to peek ahead). It is also very important to remember that the main benefit you get from the activities is due to the time and effort you have spent on them. It doesn't really matter so much whether you have actually found all the right answers. We repeat that the main role for the activities is to create an experiential basis, an intuitive familiarity with the mathematical ideas. The right answers will come after you have read the text, or after you have discussed matters in class and with your colleagues. In a few cases, however, there are some "right answers" that represent very deep ideas and you may not get them for a long time. Another part of learning mathematics is to learn to live with ideas that you only understand partially, or not at all. Through activities and discussion, you will come to understand more and more aspects of a topic. Eventually, it will all fit together and you will begin to understand the subject as a whole.

After the discussion in a section come the exercises. These are fairly stan­dard because they come after you have had every opportunity to construct the mathematics in the section. The purpose of the exercises is to help

Comments for the Student xv

you solidify your knowledge, to challenge your thinking, and to give you a chance to relate to some mathematical ideas that were not included in the section. You will find no cases in which the text lays out a mechani­cal procedure for solving a certain class of problems and then asks you to apply this procedure mindlessly to solve numerous problems of the same class. We omit this kind of material because, as must be clear to you by now, we believe that sort of interaction is of little help in real learning of mathematics.

Last words of wisdom

We have had our say, now its your turn to roll up your sleeves and start working. We hope that as you struggle through the activities, the text and the exercises, and especially as you struggle with the frustration which in­evitably must accompany any meaningful learning of such deep material, you will not lose the Grand View of what you are doing-namely, learning successfuly and meaningfully one of the most beautiful albeit difficult pieces of mathematics. Especially we hope that you will succeed in maintaining an attitude of play, exploration and wonderment, which is what the spirit of true mathematics is all about.

Ed Dubinsky Uri Leron

January, 1993

Comments for the Instructor

Teaching a course with this book

This book is intended to support a constructivist (in the epistemological, not mathematical sense) approach to teaching. That is, it can be used in an undergraduate abstract algebra course to help create an environment in which students construct, for themselves, mathematical concepts appro­priate to understanding and solving problems in this area. Of course, the pedagogical ideas on which the book is based do not appear explicitly in the. text, but rather are implicit in the structure and content.

In a sense this book lies somewhere in between a traditional text that supports a lecture method of teaching, and a book such as Halmos' A Hilbert Space Problem Book that can support a Moore-style approach. The ideas in our text are not presented in a completed, polished form, adhering to a strict logical sequence, but roughly and circularly, with the student responsible for eventually straightening things out. On the other hand, the student is given considerable help in making mathematical constructions to use in making sense out of the material. This help comes from a combina­tion of computer activities, leading questions and a conversational style of writing. It should be noted that although it is assumed that each learning cycle begins with activities, the students are not expected to discover all the mathematics for themselves. In fact, since the main purpose of the ac­tivities is to establish an experiential basis for subsequent learning, anyone who spends a considerable time and effort working on them, will reap the benefits whether they have discovered the "right" answers or not.

xviii Comments for the Instructor

It is also important to point out that the book is not primarily intended as a reference. Our main concern has been writing a book that will best facilitate a student's first introduction into abstract algebra. We see nothing wrong if, after learning the material through a course based on this book, a student uses some traditional text that is more suitable as a reference for someone who has already been through a first introduction.

In teaching abstract algebra courses based on this book, we are find­ing that our approach appears to be extremely effective for most students (omitting, perhaps, the top and bottom 5% ability group), bringing them much more into an understanding of the ideas in this subject than one would think possible from the usual experience with this course. For the superior student, the exercise set is strong enough to challenge and whet an appetite for more advanced mathematics. For all levels, we find that the students who go through our course develop a more positive attitude towards mathematical abstraction and mathematics in general.

Finally, before describing the structure of the book, we should point out that the text is only part of the course. We have available a package to aid instructors in using our approach. This package includes: a disk with the required software (running on Macintosh or PC); documentation for the software; complete sets of assignments, class lesson plans, and sample exams; answer keys; and information on dividing students into teams. We have also been experimenting with an alternative method for introducing students to ISETL more quickly. This method, which may be suitable for some classes, utilizes four worksheets and discussions based on them, as a substitute for the more extensive Chapter 1. These worksheets are also included in the package for instructors. Please contact the authors for more information.

The ACE cycle

The text is divided into sections, each intended to be covered by an average class in about a week. Each section consists of a set of activities, class discussion material, and a set of exercises.

Activities. These are tasks that present problems which require students to write computer code in ISETL representing mathematical con­structs that can be used to solve the problems. Often, an activity will require use of mathematics not yet covered in the text. The student is expected to discover the mathematics or even just make guesses, possibly reading ahead in the text for clues or explanations.

Class Discussion Material. These portions contain some explanations, some completed mathematics and many questions, all taking place under the assumption that the student has already spent consider­able time and effort on the activities related to the same topics. Our

Comments for the Instructor xix

experience indicates that with this background, students can relate much more meaningfully to the formal definitions and theorems. Each unanswered question in the text is either answered later in the book or repeated as an explicit problem in the exercises. Our way of us­ing this discussion material in a course is to have the students in a class working together in teams to solve paper and pencil problems, mainly suggested by the open questions in the text. This largely re­places lectures which occur only as summaries after the students have had a chance, through the activities and discussions, to understand the material.

Exercises. These are relatively traditional and are used to reinforce the ideas that the students have constructed up to this point. They oc­casionally introduce preliminary versions of topics that will be con­sidered later.

Covering the course material

Though the teaching method supported by this book is novel, the selection of material is standard. The book contains the material on groups, rings and fields usually covered in a one-semester course, though we would be happier if we could stretch it over 1.5 or 2 semesters. We feel that for many students, going beyond the material on group theory in one semester interferes with their ability to advance beyond a superficial understanding of Abstract Algebra.

The first chapter covers all the necessary knowledge and practice on ISETL, while at the same time introducing some of the mathematical systems (such as modular arithmetic and permutations) that will be used most often in the remainder of the book. Chapter 2 introduces the group concept, while Chapters 3 and 4 take the student deeper into group theory, notably Lagrange's theorem and the fundamental theorem of homomor­phisms. Chapter 5 introduces ring theory, always building on the analogy between groups and rings. Finally, Chapter 6 brings the theory of factor­ization in integral domains, presented as a generalization and abstraction of the well-known facts about the integers and about polynomials. It cul­minates in the fundamental theorem of field theory about the possibility of adjoining roots of polynomials over fields.

Ed Dubinsky Uri Leron

January, 1993

Acknow ledgments

The authors are grateful to the following people who read earlier versions of the book and made valuable comments:

David Chillag, Orit Hazzan, Robert Smith, and Alfred Tang.

We would also like to acknowledge the contributions to our work which have been made by the members of our Abstract Algebra Project: Jennie Dautermann, James Kaput, Robert Smith, and Rina Zazkis. This project has been supported by a grant from the National Science Foundation and we would like to express our gratitude for that.

Finally, we would like to thank the many students who have been willing to subject themselves to the very unusual kind of undergraduate mathe­matical experience supported by this text. We hope that their growth in understanding of the beautiful subject of abstract algebra has justified their hard work and their courage to plunge into the unknown.